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A227527 G.f.: Sum_{n>=0} x^n * (1-x)^A003188(n), where A003188(n) = n XOR [n/2] is the Gray code for n.

%I A227527 #10 Jul 14 2013 14:36:12
%S A227527 1,1,0,-2,2,-5,9,-3,-13,14,35,-149,300,-450,673,-1151,1856,-2366,2424,
%T A227527 -3192,9319,-32687,96858,-238410,508290,-998065,1925065,-3750685,
%U A227527 7162328,-12635545,19437562,-24482990,22154946,-6283107,-25823457,69598597,-113006459,140737910,-137007218,67953174
%N A227527 G.f.: Sum_{n>=0} x^n * (1-x)^A003188(n), where A003188(n) = n XOR [n/2] is the Gray code for n.
%C A227527 A(1/2) = Sum_{n>=0} 1/2^(n + A003188(n)) = 1.3146990537656455533266364007421...
%H A227527 Paul D. Hanna, <a href="/A227527/b227527.txt">Table of n, a(n) for n = 0..1024</a>
%F A227527 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(A003188(k), n-k), where A003188(k) = k XOR [k/2].
%e A227527 G.f.: A(x) = 1 + x - 2*x^3 + 2*x^4 - 5*x^5 + 9*x^6 - 3*x^7 - 13*x^8 + 14*x^9 +...
%e A227527 where A(x) = A(1-x) equals the series:
%e A227527 A(x) = 1 + x*(1-x) + x^2*(1-x)^3 + x^3*(1-x)^2 + x^4*(1-x)^6 + x^5*(1-x)^7 + x^6*(1-x)^5 + x^7*(1-x)^4 + x^8*(1-x)^12 + x^9*(1-x)^13 + x^10*(1-x)^15 + x^11*(1-x)^14 + x^12*(1-x)^10 + x^13*(1-x)^11 + x^14*(1-x)^9 + x^15*(1-x)^8 +...
%o A227527 (PARI) {a(n)=polcoeff(sum(m=0, n, x^m*(1-x+x*O(x^n))^bitxor(m,m\2)), n)}
%o A227527 for(n=0, 64, print1(a(n), ", "))
%o A227527 (PARI) {a(n) = sum(k=0, n, (-1)^(n-k)*binomial(bitxor(k,k\2), n-k))}
%o A227527 for(n=0, 64, print1(a(n), ", "))
%Y A227527 Cf. A227526, A003188.
%K A227527 sign
%O A227527 0,4
%A A227527 _Paul D. Hanna_, Jul 14 2013